About advanced algorithm of statistical simulation of seismic noise in the flat observation area for determination the frequency characteristics of geological environment

(Рекомендовано членом редакційної колегії д-ром фіз-мат. наук, проф. Б.П. Масловим) The article is devoted to the theory and methods of random process and field statistical simulation on the basis of their spectral decomposition and modified Kotelnikov-Shennon interpolation sums, as well as using these methods in environmental geophysical monitoring. The problem of statistical simulation of the 3D random fields (homogeneous in time and homogeneous isotropic with respect to the 2 other variables) is considered for introducing into seismological researches for determination the frequency characteristics of geological environment. Statistical model and advanced numerical algorithm of simulation realizations of such random fields are built on the basis of modified interpolation Kotelnikov-Shennon decompositions for generating the adequate realizations of seismic noise. Real-valued random fields ξ(t, x), t R, x R, that are homogeneous with respect to time and homogeneous isotropic with respect to spatial variables in the multidimensional space are studied. The problem of approximation of such random fields by random fields with a bounded spectrum is considered. An analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum is presented. Improved estimates of the mean-square approximation of random fields in the space R R by a model constructed with the help of the spectral decomposition and interpolation Kotelnikov–Shannon formula are obtained. Some procedures for the statistical simulation of realizations of Gaussian randomfieldswith a bounded spectrum that are homogeneous with respect to time and homogeneous isotropic with respect to spatialvariables in the 2D space are developed. There has been proved theorems on the mean-square approximation of homogeneous in time and homogeneous isotropic with respect to the two other variables random fields by special partial sums. A simulation method was used to formulate an advanced algorithm of numerical simulation by means of this theorem. The spectral analysis methods of generated seismic noise realizations are considered. There have been developed universal methods of statistical simulation (Monte Carlo methods) of multiparameters seismology data for generating seismic noise on 2D grids of required detail and regularity.